Internal coordinate phase space

Propagate by harmonic part of Hq for the time Arjl. This corresponds to the rotation of internal normal coordinates, P( and Q[, in the phase space by the corresponding vibrational frequency Ui... 

Fig. x.4. Schematic representation of the motion in phase space of the internal coordinates of a critically energized molecule with a single possible mode of decomposition. Bounding surface is one of constant potential energy. 

Hulburt and Katz (HI7) developed a framework for the analysis of particulate systems with the population balance equation for a multivariate particle number density. This number density is defined over phase space which is characterized by a vector of the least number of independent coordinates attached to a particle distribution that allow complete description of the properties of the distribution. Phase space is composed of three external particle coordinates x and m internal particle coordinates Xj. The former (Xei, x 2, A es) refer to the spatial distribution of particles. The latter coordinate properties Ocu,Xa,. . , Xt ) give a quantitative description of the state of an individual particle, such as its mass, concentration, temperature, age, etc. In the case of a homogeneous dispersion such as in a well-mixed vessel the external coordinates are unnecessary whereas for a nonideal stirred vessel or tubular configuration they may be needed. Thus (x t)d represents the number of particles per unit volume of dispersion at time t in the incremental range x, x -I- d, where x represents both coordinate sets. The number density continuity equation in particle phase space is shown to be (HI 8, R6)... 

In this section the population balance modeling approach established by Randolph [95], Randolph and Larson [96], Himmelblau and Bischoff [35], and Ramkrishna [93, 94] is outlined. The population balance model is considered a concept for describing the evolution of populations of countable entities like bubble, drops and particles. In particular, in multiphase reactive flow the dispersed phase is treated as a population of particles distributed not only in physical space (i.e., in the ambient continuous phase) but also in an abstract property space [37, 95]. In the terminology of Hulburt and Katz [37], one refers to the spatial coordinates as external coordinates and the property coordinates as internal coordinates. The joint space of internal and external coordinates is referred to as the particle phase space. In this case the quantity of basic interest is a density function like the average number of particles per unit volume of the particle state space. The population balance may thus be considered an equation for the number density and regarded as a number balance for particles of a particular state. 

The disperse phase is constituted by discrete elements. One of the main assumptions of our analysis is that the characteristic length scales of the elements are smaller than the characteristic length scale of the variation of properties of interest (i.e. chemical species concentration, temperature, continuous phase velocities). If this hypothesis holds, the particulate system can be described by a continuum or mean-field theory. Each element of the disperse phase is generally identified by a number of properties known as coordinates. Two elements are identical if they have identical values for their coordinates, otherwise elements are indistinguishable. Usually coordinates are classified as internal and external. External coordinates are spatial coordinates in fact, the position of the elements in physical space is not an internal property of the elements. Internal coordinates refer to more intimate properties of the elements such as their momenta (or velocities), their enthalpy... 

It is straightforward that the quantity ( )d represents the number density of disperse entities contained in the phase-space volume d centered at per unit of physical volume. If we integrate the NDF over all possible values of the internal-coordinate vector we obtain the total number concentration N(t, x) ... 

A special case of considerable interest occurs when the internal-coordinate vector is the particle-velocity vector, which we will denote by the phase-space variable v. In fact, particle velocity is a special internal coordinate since it is related to particle position (i.e. external coordinates) through Newton s law, and therefore a special treatment is necessary. We will come back to this aspect later, but for the time being let us imagine that otherwise identical particles are moving with velocities that may be different from particle to particle (and different from the surrounding fluid velocity). It is therefore possible to define a velocity-based NDF nv(t, x, v) that is parameterized by the velocity components V = (vi, V2, V3). In order to obtain the total number concentration (i.e. number of particles per unit volume) it is sufficient to integrate over all possible values of particle velocity Oy ... 

Equation (2.14) must be coupled with initial conditions given for the starting time and with boundaries conditions in physical space O and in phase space O. Analytical solutions to Eq. (2.14) are available for a few special cases and only under conditions specified by some very simple hypotheses. However, numerical methods can be used to solve this equation and will be presented in Chapters 7 and 8. The numerical solution of Eq. (2.14) provides knowledge of the NDE for each time instant and at every physical point in the computational domain, as well as at every point in phase space. As has already been mentioned, sometimes the population of particles is described by just one internal coordinate, for example particle length (i.e. f = L), and the PBE is said to be univariate. When two internal coordinates are needed, for example particle volume and surface area (i.e. = (v, a)), the PBE is said to be bivariate. More generally, higher-dimensional cases are referred to as multivariate PBEs. Another important case occurs when part of the internal-coordinate vector is equal to the particle-velocity vector (i.e. when the particles are characterized not by a unique velocity field but by their own velocity distribution). In that case, the PBE becomes the GPBE, as described next. 

When performing the Reynolds averaging, the internal-coordinate vector is not affected. Thus, the Reynolds average commutes with time, space, and phase-space gradients. See Fox (2003) for a discussion of this topic. 

The last property ensures that all the lower-order mixed moments are included in the set. This turns out to be very important since generally neglecting mixed moments leads to abscissas that lie on lower-dimensional subspaces of the M-dimensional phase space. Whenever the dynamics of the investigated problem is not confining the abscissas on lower-dimensional supports, it is appropriate to choose a moment set to define A that is not restricted to generating such behavior. Therefore, it is suggested for many applications that one should use moment sets that treat all the M internal coordinates equally. If we relax this third condition a valid (but not optimal) moment set is instead obtained. 

As in Exercise 3.6, these four equations can easily be found by the observation of the position of the eight nodes in the phase space of the three internal coordinates (see Figure 3.2). The additional four equations are obtained by fixing some mixed moments (i.e. below we show the equations for the most natural choice affixing the lowest-order moments possible mi,0,1, mo,i,i, an[Pg.72]

Note that for the third internal coordinate the quadrature nodes are calculated from each of the N N2 nodes of Figure 3.4, along vertical lines perpendicular to the plane generated in phase space by and 2, resulting in A iA 2A 3 nodes and N N2N weights. This corresponds to the following functional assumption for the NDF ... 

Note that left-hand side of this expression is, in fact, a continuity equation for which states that the multi-particle joint PDF is constant along trajectories in phase space. The term on the right-hand side of Fq. (4.32) has a contribution due to the Alp-particle collision operator, which generates discontinuous changes in particle velocities Up" and internal coordinates p", and to particle nucleation or evaporation. The first term on the left-hand side is accumulation of The remaining terms on the left-hand side represent... 

The first term on the right-hand side of this equation is a total derivative, so it can be integrated formally. However, for internal coordinates the phase space does not usually extend to infinity. In order to see clearly what can happen, consider the case with a single internal coordinate that is bounded by zero and infinity ... 

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